Part 1: The Total Derivative

To this point, we have considered only partial derivatives of a function f( x,y) . In this section, we introduce the concept of a total derivative of a transformation.  

Recall that a function f(x) of one variable is said to be differentiable at p if the following limit exists:

f' (p) =   lim h ® 0     f(p+h) - f( p) h

(1)

The problem with extending (1) to higher dimensions is that it requires h to become a vector, and yet we cannot "divide" by a vector.  However, we can rewrite (1) in the form

lim h ® 0   f(p+h) - f( p) h   -  f' (p)    =  0

and thus, we obtain also that

lim h ® 0   f(p+h) - f( p) h   -    f' (p) h h   =  0

Consequently, the definition of the derivative (1) can be re-interpreted to mean that if f(x) is differentiable at p if there is a number f'(p) such that

lim h ® 0   | f(p+h) - f( p)  -  f' (p) h | |h|   =  0

In this formulation, division is by |h|, which naturally generalizes to the norm of a vector h = á h, k ñ. Indeed, if we denote f( x,y) by f(x) , where x = á x,y ñ, and let p = ( p,q) be a point, then we have the following:   

Definition 5.1: A function f( x,y) is differentiable at a point ( p,q) if there exists a vector Ñf( p) for which

lim h® 0     | f(p+h) -  f(p) - Ñ f( p) · h| || h|| = 0

(2)

When it exists, Ñf( p) is the total derivative of f( x) at p.

The vector Ñf( p) is also called the gradient of f( x) at p.

If we write Ñf( p) = á a,b ñ , then we can determine the values of a and b by evaluating the limit in (2) in two different directions. Along the x-axis, h = á h,0 ñ and thus we have

lim h® 0     | f(p+h,q) -  f(p,q) - á a,b ñ · á h,0 ñ| || á h,0 ñ || = 0

lim h ® 0   f(p+h,q) - f( p,q)  -  ah h   =  0

lim h ® 0   f(p+h,q) - f( p,q) h   -    a   =  0

This implies that

a =   lim h ® 0     f(p+h) - f( p) h   =  fx(p,q)

(1)

Similarly, (see the exercises), it can be shown that b = f_y(p,q), so that the gradient is given by

Ñf( p,q) = á fx( p,q) ,fy(p,q) ñ

This yields the following:    

Theorem 5.2: If f( x,y) is differentiable at a point ( p,q) , then its total derivative is given by the gradient of f at ( p,q) , which is

Ñf( p,q) = á fx( p,q),  fy(p,q) ñ

The total derivative is also known as the Jacobian of the mapping f( x,y) for reasons that will become apparent in the next chapter. 

EXAMPLE 1    Find the total derivative (i.e., gradient) of

f( x,y) = x2+3xy

Solution:  Since f_x = 2x+3y and f_y = 3x, the total derivative is

Ñf = á 2x+3y,3x ñ

Definition 5.1 can be applied to a function f of any number of variables, in which case Theorem 5.2 says that Ñf is the vector of first partial derivatives. For example, the gradient of a function of 3 variables U( x,y,z) is given by

ÑU = á Ux,Uy,Uz ñ

and ÑU is the total derivative of U( x,y,z) .

  Check your Reading: Why was the product rule not used in evaluating  f_x(x,y) in example 1?